Forward Error Correction

1
Forward Error Correction

• Steven Marx
• CSC457
• 12/04/2001

2
Outline

• What is FEC?
• Why do we need it?
• How does it work?
• Where is it used?

3
What is FEC?

• Send k packets
• Reconstruct n packets
• Such that we can tolerate k-n losses
• Called an (n, k) FEC code

4
What is FEC? (2)
5
Why FEC?

• Alternatives
• ARQ (Automatic Repeat reQuest)
• requires feedback
• bad for multicast
• tolerance
• only suitable for some applications

6
Why FEC? (2)

• Advantages
• sometimes no feedback channel necessary
• long delay path
• one-way transmission
• avoids multicast problems
• Disadvantages
• computationally expensive
• requires over-transmission

7
How is this possible?
An easy example (n, k) (2, 3) FEC
code transmitting two numbers a and b Send
three packets 1. a 2. b 3. a b
8
How is this possible? (2)
Could be represented as matrix multiplication To
encode
To decode, use subset of rows.
9
How is this possible? (3)
More generally y Gx, where G is a generator
matrix G is constructed in such a way that any
subset of rows is linearly independent. A
systematic generator matrix includes the
identity matrix.
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A Problem

• a and b are 8-bit numbers
• a b may require more bits
• loss of precision means loss of data

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A Solution

• Finite fields
• field
• we can add, subtract, multiply, and divide as
  with integers
• closed over these operations
• finite finite number of elements

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A Solution (2)

• Specific example
• prime field or Galois Field - GF(p)
• elements 0 to p-1
• modulo p arithmeticProblem
• with the exception of p 2, ?log(p)?gt log(p)
  bits required
• requires modulo operations

13
Extension Fields

• q pr elements with p prime, r gt 1
• extension field, or GF(pr)
• elements can be considered polynomials of
  degree r - 1
• sum just sum modulo p
• extra simple with p 2
• exactly r bits needed
• sums and differences just XORs

14
Multiplication and Division

• Exists an a whose powers generate all non-zero
  elements.
• In GF(5), a 2, whose powers are (1,2,4,3,1,).
• Powers of a repeat with period q - 1, so aq-1
  a0 1

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Multiplication and Division (2)

• for all x, x al
• l is xs logarithm

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Multiplication and Division (3)
An example GF(5) -gt a 2 3 23 mod 5 4 22
mod 5 3 4 232 mod 5 32 mod 5
2 mod 5 3 4 12 mod 5 2 mod 5
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Vandermonde Matrices

• gi,j xij-1
• xis are elements of GF(pr)
• called Vandermonde Matrices
• invertible if all xis different
• y Gx
• G-1y G-1Gx x
• can be extended with the identity matrix for
  systematic codes

18
Swarmcast - a real example

• for media distribution
• reduces bandwidth requirements of the server
• server transmits to a small number of clients
• while downloading, those clients also transmit
  packets to other clients
• FEC used to maximize chances of getting useful
  packets

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Swarmcast (2)
Star Wars Episode Two Trailer
300Mb/s
100Mb/s
100Mb/s
50Mb/s
100Mb/s
100Mb/s
50Mb/s
50Mb/s
100Mb/s
100Mb/s
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Other useful applications

• multicast
• streaming media less I frames in MPEGS
• one-way communication
• high delay pathways
• storage

21
Conclusion

• FEC
• allows error correction without retransmission
• requires redundancy in transmission
• useful for multicast
• not extensively used at the packet level
• more important with high bandwidth, high
  latency, as is the trend